## Energy and mass

## Dependence of nuclear radius on nucleon number

What is the relationship between the radius of a nucleus, *R*, and the number of nucleons in the nucleus , *A* (AKA the mass number, *N*)?

Use the following data to investigate this. Assuming it is a power relationship, recall that we can find the log of both sides in order to discover what this power is.

Nucleon number, A |
Nuclear radius, R (fm) |

7 | 2.30 |

14 | 2.89 |

31 | 3.77 |

88 | 5.34 |

120 | 5.92 |

157 | 6.47 |

197 | 6.98 |

239 | 7.45 |

## Radioactive Decay Equations

** emission**

** emission**

** emission**

**Electron capture**

** emission**

## Butter Guns and Inverse Square Laws

A *butter gun* was designed by a restaurant in order to butter pieces of bread very quickly. Charged with one knob of butter (its maximum capacity, which it fired every time) it could butter one piece of bread effectively at a distance of 1 foot.

However, one day someone discovered that if the gun was fired from 2 feet instead, it could butter *four* pieces of bread. The restaurant manager was very excited, as this meant his staff were that much more efficient.

Whatsmore, it was discovered that if the gun was fired from 3 feet, it could cover an impressive *nine* pieces of bread. Needless to say, the manager was pleased. However, the customers soon started complaining.

Why were the customers complaining?

If one knob of butter gets on one piece of bread from 1 foot away, how much is on each piece of bread at 2 feet?

How much is on each piece of bread at 3 feet?

Can you see the pattern; how many pieces of bread could be covered at 7 feet, and how much butter would be on each piece of bread?

## Radioactive Decay Equations

When atomic nuclei become *unstable*, they must emit either an α or β particle, or energy in the form of a γ ray. They become more stable in the process.

*A*= activity of a sample, Bq (Becquerels) = the rate of decay = the number of nuclei that will decay in any given second*N*= the number of unstable nuclei remaining in a sample*λ*= the probability that any given nucleus will decay in any given second- = the half-life of a sample, s (seconds)

If *λ* is the probability that any given nucleus will decay in a second, and there are *N* nuclei remaining, then it stands to reason that *λN* gives us the number of nuclei that will decay in a given second;

The number of nuclei remaining after some time *Δt* is *reduced* (hence the minus sign) by *ΔN*. Then, the activity of the sample is:

Or,

This is a *linear differential equation* whose solution is:

where

= the number of undecayed nuclei at time zero

*N* = the number of undecayed nuclei at time t

Thus, radioactive decay follows an exponential decay curve.

Now prove that

and

## Radioactive Dice (Advanced)

This experiment investigates the random nature of radioactivity by studying the nature of another random process, rolling several dice at once. After each roll a certain number is removed (based on the ‘instability’ of the atoms we are modelling). The decay pattern follows an exponential decline, given by

Taking logs of both sides we find:

This means that if we plot a graph of *ln|N|* against *t*, we should find a straight line with gradient *λ*.

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